On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

2021/03/17 by Damelin, Steven B. · 1 citation
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paper · doi:10.48550/arxiv.2103.09748

Abstract

This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensional data science. Modern techniques in algebraic geometry, approximation theory, computational harmonic analysis and extensions develop the first of its kind, a unified framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in of \mathbb R^D with the near isometry extension problem for discrete and non-discrete subsets of \mathbb R^D with certain geometries. In addition, the paper surveys related work on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems are given.

Cited by

• A Bounded mean oscillation (BMO) theorem for small distorted diffeomorphisms from \mathbb R^D to \mathbb R^D and PDE

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